This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A355327 #14 Jul 01 2022 12:20:29
%S A355327 1,1,5,10,39,83,317,678,2585,5531,21085,45116,171987,368005,1402873,
%T A355327 3001764,11443033,24484957,93339173,199720270,761354199,1629089495,
%U A355327 6210256613,13288248522,50656169297,108390330503
%N A355327 Number of ways to tile a 2 X n board with squares and dominoes where vertical dominoes are only allowed in even-numbered locations.
%C A355327 Similar in spirit to A030186, which counts all tilings of a 2 X n board without any restrictions on locations of vertical dominoes.
%H A355327 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,9,0,-7,0,1).
%F A355327 a(2*n-1) = Sum_{k=1..2*n-1} k*a(2*n-1-k).
%F A355327 a(2*n-1) = a(2*n-2) + 4*a(2n-3) + a(2*n-4) - a(2*n-5).
%F A355327 a(2*n) = 2*a(2*n-1) + 4*a(2n-2) - a(2*n-4).
%F A355327 G.f.: (1 + 3*x + x^2)*(1 - x)^2/(1 - 9*x^2 + 7*x^4 - x^6).
%F A355327 a(n) = 9*a(n-2) - 7*a(n-4) + a(n-6).
%e A355327 This is one of the a(4)=39 possible tilings of a 2 X 4 board. Note that vertical dominoes can only occur in the second or fourth location (we have one vertical domino in the second location in this picture).
%e A355327 _______
%e A355327 |_| |___|
%e A355327 |_|_|_|_|
%t A355327 LinearRecurrence[{0, 9, 0, -7, 0, 1}, {1, 1, 5, 10, 39, 83}, 20]
%Y A355327 Cf. A030186.
%K A355327 nonn,easy
%O A355327 0,3
%A A355327 _Greg Dresden_ and _Zijie He_, Jun 28 2022